The Inverse Problem for a Quasilinear Wave Equation with Memory

Abstract

The forward and inverse problems are investigated for the quasilinear wave equation \( \square u -q(x)u^{2}-K\ast u=0 \) where the kernel \( K(x,t) \) is represented in the form \( K(x,t)=p(x) K_0(t) \) with \( p(x) \) being a continuous function. The inverse problem is devoted to the determination of the compact functions \( q(x) \) and \( p(x) \). Traces of the derivative with respect to \( x \) of two solutions to the forward initial–boundary value problem related to two arbitrary boundary data are given for \( x=0 \) on the finite segment \( [0,T] \) as an additional information for the solution to the inverse problem. The conditions for the unique solvability of the forward problem are found. A local existence and uniqueness theorem is proved for the inverse problem.